ANOSOV REPRESENTATIONS AND PROPER ACTIONS FRANÇOIS GUÉRITAUD, OLIVIER GUICHARD, FANNY KASSEL, AND ANNA WIENHARD Abstract. We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan pro- jection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups. 1. Introduction Anosov representations of word hyperbolic groups into real Lie groups were first in- troduced by Labourie [Lab06]. They provide an interesting class of discrete subgroups of semisimple or reductive Lie groups, with a rich structure theory. In many respects they generalize, to a higher-rank setting, the convex cocompact representations into rank-one simple groups [GW12, KLPa, KLPb, KLPc]. They also play an important role in the context of higher Teichmüller spaces. The original definition of Anosov representations from [Lab06] involves the flow space of a word hyperbolic group, whose construction is not completely straightfor- ward. In this paper, we establish several characterizations of Anosov representations that do not involve the flow space. A central role in our characterizations is played by the Cartan projection of G (associated with a fixed Cartan decomposition), which measures dynamical properties of diverging sequences in G. We apply our characteri- zations to the study of proper actions on homogeneous spaces by establishing a direct link between the properties, for a representation ρ :Γ ! G to be Anosov, and for Γ to act properly discontinuously via ρ on certain homogeneous spaces of G. We now describe our results in more detail. 1.1. Existence of continuous boundary maps. Since the foundational work of Furstenberg and the celebrated rigidity theorems of Mostow and Margulis, the ex- istence of boundary maps has been playing a crucial role in the study of discrete subgroups of Lie groups. Given a representation ρ :Γ ! G of a finitely generated group Γ into a reductive (e.g. semisimple) Lie group G, measurable ρ-equivariant boundary maps from a Poisson boundary of Γ to a boundary of G exist under rather weak assumptions, e.g. Zariski density of ρ(Γ) in G. However, if Γ comes with some FG, OG, and FK were partially supported by the Agence Nationale de la Recherche under the grants ETTT (ANR-09-BLAN-0116-01) and DiscGroup (ANR-11-BS01-013), as well as through the Labex CEMPI (ANR-11-LABX-0007-01). OG also received funding from the European Research Council under the European Community’s seventh Framework Programme (FP7/2007-2013)/ERC grant agreement FP7-246918. AW was partially supported by the National Science Foundation under agreements DMS-1065919 and 0846408, by the Sloan Foundation, by the Deutsche Forschungsge- meinschaft, by the European Research Council under ERC-Consolidator grant 614733, and by the Klaus Tschira Foundation. This paper was completed while OG, FK, and AW were in residence at the MSRI in Berkeley, California, supported by the National Science Foundation under grant 0932078 000. 1 ANOSOV REPRESENTATIONS AND PROPER ACTIONS 2 geometric or topological boundary, obtaining a continuous ρ-equivariant boundary map is in general difficult. Anosov representations of word hyperbolic groups come, by definition, with a pair of continuous equivariant boundary maps. More precisely, let ρ :Γ ! G be a Pθ-Anosov representation. (We always assume G to be noncompact and linear, and use the − notation Pθ;Pθ for its standard parabolic subgroups with the convention that P; = G, + see Section 2.2.) Then there exist ρ-equivariant boundary maps ξ : @1Γ ! G=Pθ − − and ξ : @1Γ ! G=Pθ that are continuous. These boundary maps have additional 0 remarkable properties: they are transverse, i.e. for any distinct points η; η 2 @1Γ + − 0 − the images ξ (η) 2 G=Pθ and ξ (η ) 2 G=Pθ are in general position, and they are dynamics-preserving, which means that for any γ 2 Γ of infinite order with attracting + + + − + fixed point ηγ 2 @1Γ, the point ξ (ηγ ) (resp. ξ (ηγ )) is an attracting fixed point − for the action of ρ(γ) on G=Pθ (resp. G=Pθ ). Furthermore, these maps satisfy an exponential contraction property involving certain bundles over the flow space of Γ (see Section 2.5). In this paper, given a word hyperbolic group Γ and a representation ρ :Γ ! G, we construct, under some growth assumption for the Cartan projection of G restricted to ρ(Γ) (Theorem 1.1.(1)), an explicit pair (ξ+; ξ−) of continuous, ρ-equivariant bound- ary maps. (Recall that the Cartan projection µ : G ! a+, defined from a Cartan decomposition G = K(exp a+)K, is a continuous, proper, surjective map to the closed + Weyl chamber a ; see Section 2.3.1, and Example 2.16 for G = GLd(R).) We give a sufficient condition for these maps to be dynamics-preserving (Theorem 1.1.(2)). Under an additional assumption on the growth of µ along geodesic rays in Γ (The- orem 1.1.(3)), we prove that the pair of maps (ξ+; ξ−) is also transverse and that ρ is Anosov. This assumption involves the following notion: we say that a sequence N 0 00 000 (xn) 2 (R+) is CLI (i.e. has coarsely linear increments) if there exist κ, κ ; κ ; κ > 0 such that for all n; m 2 N, 0 00 000 (1.1) κm − κ ≤ xn+m − xn ≤ κ m + κ : In other words, n 7! xn is a quasi-isometric embedding of N into R+. Theorem 1.1. Let Γ be a word hyperbolic group and j · jΓ :Γ ! N its word length function with respect to some fixed finite generating subset of Γ. Let G be a real reductive Lie group and ρ :Γ ! G a representation. Fix a nonempty subset θ ⊂ ∆ + ∗ of the simple restricted roots of G (see Section 2.2.2), and let Σθ ⊂ a be the set of positive roots that do not belong to the span of ∆ r θ. (1) If there is a constant C > 0 such that for any α 2 θ, hα; µ(ρ(γ))i ≥ 2 log jγjΓ − C; + then there exist continuous, ρ-equivariant boundary maps ξ : @1Γ ! G=Pθ − − and ξ : @1Γ ! G=Pθ . (2) If moreover for any α 2 θ and any γ 2 Γ, hα; µ(ρ(γn))i − 2 log jnj −! +1; jnj!+1 then ξ+ and ξ− are dynamics-preserving. + (3) If moreover for any α 2 Σθ and any geodesic ray (γn)n2N in the Cayley graph + − of Γ, the sequence hα; µ(ρ(γn))i n2N is CLI, then ξ and ξ are transverse and ρ is Pθ-Anosov. ANOSOV REPRESENTATIONS AND PROPER ACTIONS 3 Let us briefly discuss the meaning of the assumptions of Theorem 1.1. Let k · k be a Euclidean norm on the Cartan subspace a which is invariant under the restricted + Weyl group of a in G. For any α 2 ∆, the function hα; ·i : a ! R+ is proportional to the distance function to the wall Ker(α), with respect to k·k. Thus the assumption of Theorem 1.1.(1) means that the set µ(ρ(Γ)) “drifts away at infinity” from the union S + of walls α2θ Ker(α) in a , at speed at least 2 log in the word length. The CLI assumption in Theorem 1.1.(3) means that the image under µ ◦ ρ of any geodesic ray in the Cayley graph of Γ drifts away “forever linearly” from the hyperplane + Ker(α) for every α 2 Σθ (see Section 2.3.2). Inside the Riemannian symmetric space G=K of G, the function kµ(·)k : G ! R+ gives the distance between the basepoint x0 = eK 2 G=K and its image under an element of G (see (2.8)). The functions hα; µ(·)i can be thought of as refinements of kµ(·)k. The CLI assumption in Theorem 1.1.(3) means that hα; µ ◦ ρ(·)i :Γ ! R+ restricts to a quasi-isometric + embedding on any geodesic ray in the Cayley graph of Γ, for α 2 Σθ . Remarks 1.2. (a) Theorem 1.1.(1) extends a rank-one result of Floyd [Flo80] to the setting of higher real rank. As in [Flo80], the image of ξ+ in Theorem 1.1 is the limit set of Γ in G=Pθ (Definition 5.1), see Theorem 5.3. (b) Boundary maps are commonly constructed only on some large subset of @1Γ (dense or of full measure). By contrast, Theorem 1.1.(1) is based on an explicit construction of the boundary maps at every point: see Theorem 5.3. (c) Theorem 5.3 refines Theorem 1.1 by providing weaker conditions for the existence of ρ-equivariant boundary maps with various properties. The sub- tleties in dropping one assumption among continuity, dynamics-preservation, or transversality are illustrated by Examples 5.5 and A.6. We expect that the methods of the proof of Theorem 5.3 will have applications in broader contexts where Γ is not necessarily word hyperbolic. (d) The growth condition in Theorem 1.1.(2) is optimal (see Lemma 2.27 and Remark 2.32.(a)). (e) In Theorem 1.1.(3) the CLI constants are not required to be uniform. (f) The CLI assumption in Theorem 1.1.(3) can be restricted to the set of geodesic rays starting at the identity element e 2 Γ. In fact, we only need the CLI assumption for one quasi-geodesic representative per point in the boundary at infinity @1Γ (see Proposition 5.12). If θ = ∆ (i.e. Pθ is a minimal parabolic + subgroup of G), then the CLI assumption for all α 2 Σθ is equivalent to the + CLI assumption for all α 2 θ, because in this particular case Σθ ⊂ span(θ).